A paradox concerning the temperature distribution of a gas in a gravitational field

On the barometric formula

“…͑18͔͒ is obeyed at all heights, that is, the system is indeed isothermal. 18 Of course, this mechanical model does not correspond to true thermodynamic equilibrium ͑it is nonergodic͒. It just behaves as if it were, because a Maxwell-Boltzmann-like distribution of velocities was imposed.…”

Temperature Gradient Caused by Gravitation

“…In addition, by combining Eqs. (9) and (11) and the equation of state for ideal gas we can obtain the distribution of gas's density in a gravitational field: the higher the altitude the lower the density and the lower the temperature. In fact, the tendency of the temperature's distribution here does not influence that of the density's distribution.…”

“…Equation (11) concludes that the higher the altitude the lower the terrestrial air temperature, in accord with our daily experience such as the natural phenomenon: when spring has come and the flowers are in boom on the foot of a mountain, ice and snow still cover the top of the mountain. According to the preceding temperature gradient of −0.02 K · m −1 , the air temperature will fall by 2 K while the altitude rises by 100 m or 20 K while 1 km, approximately in accord with our practical experience.…”

Temperature gradient of vertical air column in gravitational field

“…Moreover, in this paper, Liao introduced, summarized, and criticized the papers from various viewpoints on the temperature distribution of the vertical column of an adiabatically enclosed gas in a gravitational field. Most of these papers stated that the temperature of the air column is homogeneous with no temperature gradient 12 – 15 . On the other hand, Tolman predicted that there would be a negative temperature gradient based on the theory of relativity, but its magnitude would be too small to confirm experimentally 16 – 18 .…”